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Mirrors > Home > MPE Home > Th. List > cniccbdd | Structured version Unicode version |
Description: A continuous function on a closed interval is bounded. (Contributed by Mario Carneiro, 7-Sep-2014.) |
Ref | Expression |
---|---|
cniccbdd |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 9489 |
. . 3
![]() ![]() ![]() ![]() | |
2 | ral0 3884 |
. . . 4
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3 | simp1 988 |
. . . . . . . 8
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4 | 3 | rexrd 9536 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | simp2 989 |
. . . . . . . 8
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6 | 5 | rexrd 9536 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | icc0 11451 |
. . . . . . 7
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8 | 4, 6, 7 | syl2anc 661 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 8 | biimpar 485 |
. . . . 5
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10 | 9 | raleqdv 3021 |
. . . 4
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11 | 2, 10 | mpbiri 233 |
. . 3
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12 | breq2 4396 |
. . . . 5
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13 | 12 | ralbidv 2838 |
. . . 4
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14 | 13 | rspcev 3171 |
. . 3
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15 | 1, 11, 14 | sylancr 663 |
. 2
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16 | 3 | adantr 465 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 5 | adantr 465 |
. . . . 5
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18 | simpr 461 |
. . . . 5
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19 | simp3 990 |
. . . . . . 7
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20 | abscncf 20595 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 20 | a1i 11 |
. . . . . . 7
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22 | 19, 21 | cncfco 20601 |
. . . . . 6
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23 | 22 | adantr 465 |
. . . . 5
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24 | 16, 17, 18, 23 | evthicc 21061 |
. . . 4
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25 | 24 | simpld 459 |
. . 3
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26 | cncff 20587 |
. . . . . . . 8
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27 | 22, 26 | syl 16 |
. . . . . . 7
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28 | 27 | ffvelrnda 5944 |
. . . . . 6
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29 | cncff 20587 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 19, 29 | syl 16 |
. . . . . . . . . . 11
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31 | 30 | adantr 465 |
. . . . . . . . . 10
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32 | fvco3 5869 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
33 | 31, 32 | sylan 471 |
. . . . . . . . 9
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34 | 33 | breq1d 4402 |
. . . . . . . 8
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35 | 34 | ralbidva 2836 |
. . . . . . 7
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36 | 35 | biimpd 207 |
. . . . . 6
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37 | breq2 4396 |
. . . . . . . 8
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38 | 37 | ralbidv 2838 |
. . . . . . 7
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39 | 38 | rspcev 3171 |
. . . . . 6
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40 | 28, 36, 39 | syl6an 545 |
. . . . 5
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41 | 40 | rexlimdva 2939 |
. . . 4
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42 | 41 | imp 429 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
43 | 25, 42 | syldan 470 |
. 2
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44 | 15, 43, 5, 3 | ltlecasei 9585 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4503 ax-sep 4513 ax-nul 4521 ax-pow 4570 ax-pr 4631 ax-un 6474 ax-inf2 7950 ax-cnex 9441 ax-resscn 9442 ax-1cn 9443 ax-icn 9444 ax-addcl 9445 ax-addrcl 9446 ax-mulcl 9447 ax-mulrcl 9448 ax-mulcom 9449 ax-addass 9450 ax-mulass 9451 ax-distr 9452 ax-i2m1 9453 ax-1ne0 9454 ax-1rid 9455 ax-rnegex 9456 ax-rrecex 9457 ax-cnre 9458 ax-pre-lttri 9459 ax-pre-lttrn 9460 ax-pre-ltadd 9461 ax-pre-mulgt0 9462 ax-pre-sup 9463 ax-mulf 9465 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-pss 3444 df-nul 3738 df-if 3892 df-pw 3962 df-sn 3978 df-pr 3980 df-tp 3982 df-op 3984 df-uni 4192 df-int 4229 df-iun 4273 df-iin 4274 df-br 4393 df-opab 4451 df-mpt 4452 df-tr 4486 df-eprel 4732 df-id 4736 df-po 4741 df-so 4742 df-fr 4779 df-se 4780 df-we 4781 df-ord 4822 df-on 4823 df-lim 4824 df-suc 4825 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-res 4952 df-ima 4953 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-f1 5523 df-fo 5524 df-f1o 5525 df-fv 5526 df-isom 5527 df-riota 6153 df-ov 6195 df-oprab 6196 df-mpt2 6197 df-of 6422 df-om 6579 df-1st 6679 df-2nd 6680 df-supp 6793 df-recs 6934 df-rdg 6968 df-1o 7022 df-2o 7023 df-oadd 7026 df-er 7203 df-map 7318 df-ixp 7366 df-en 7413 df-dom 7414 df-sdom 7415 df-fin 7416 df-fsupp 7724 df-fi 7764 df-sup 7794 df-oi 7827 df-card 8212 df-cda 8440 df-pnf 9523 df-mnf 9524 df-xr 9525 df-ltxr 9526 df-le 9527 df-sub 9700 df-neg 9701 df-div 10097 df-nn 10426 df-2 10483 df-3 10484 df-4 10485 df-5 10486 df-6 10487 df-7 10488 df-8 10489 df-9 10490 df-10 10491 df-n0 10683 df-z 10750 df-dec 10859 df-uz 10965 df-q 11057 df-rp 11095 df-xneg 11192 df-xadd 11193 df-xmul 11194 df-ioo 11407 df-icc 11410 df-fz 11541 df-fzo 11652 df-seq 11910 df-exp 11969 df-hash 12207 df-cj 12692 df-re 12693 df-im 12694 df-sqr 12828 df-abs 12829 df-struct 14280 df-ndx 14281 df-slot 14282 df-base 14283 df-sets 14284 df-ress 14285 df-plusg 14355 df-mulr 14356 df-starv 14357 df-sca 14358 df-vsca 14359 df-ip 14360 df-tset 14361 df-ple 14362 df-ds 14364 df-unif 14365 df-hom 14366 df-cco 14367 df-rest 14465 df-topn 14466 df-0g 14484 df-gsum 14485 df-topgen 14486 df-pt 14487 df-prds 14490 df-xrs 14544 df-qtop 14549 df-imas 14550 df-xps 14552 df-mre 14628 df-mrc 14629 df-acs 14631 df-mnd 15519 df-submnd 15569 df-mulg 15652 df-cntz 15939 df-cmn 16385 df-psmet 17920 df-xmet 17921 df-met 17922 df-bl 17923 df-mopn 17924 df-cnfld 17930 df-top 18621 df-bases 18623 df-topon 18624 df-topsp 18625 df-cn 18949 df-cnp 18950 df-cmp 19108 df-tx 19253 df-hmeo 19446 df-xms 20013 df-ms 20014 df-tms 20015 df-cncf 20572 |
This theorem is referenced by: cniccibl 21436 c1liplem1 21586 itgsubstlem 21638 cnicciblnc 28603 |
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